THE CRACK-TIP SOLUTION FOR HYDRAULIC FRACTURING IN A PERMEABLE SOLID

This paper extends previous work on self-similar analytical solutions fora hydraulically driven fracture propagating in a solid which is in a state of plane strain. In particular, we examine the effect of fluid loss to the formation modelled by the Carter leak-off mechanism. Our main new results are asymptotic solutions for arbitrary rock permeability; it is shown how these may be represented by an expansion whose leading term is the near-tip solution in the high permeability limit. This term gives an integrable singularity in the near-tip fluid pressure which is slightly stronger than the singularity which arises in the impermeable case; it follows that there is always a region immediately adjacent to the fluid front where the solution is dominated by fluid loss. Our most important conclusion for applications is that the solution in the practical case of intermediate permeability may be constructed as a series which starts from the loss dominated limit. We also provide a detailed comparison of our predictions with results from a numerical model which includes a fluid lag zone. The results are found to be in good agreement in all of the various permeability regimes.